The answer to the question in the title of this post may seem obvious (after all, isn't Bayesianism all about probabilities?), but I think that the long discussion that followed Lauren's post on van Fraassen's objection to Bayesianism from quantum mechanics shows that it isn't clear at all - or at least, that it wasn't clear to either of us as we were discussing the issue. I think that I now understand why. In this post, I'm going to give three answers to this question, which I will call The Primitivist Account (P), The Kripkean Possible Worlds Account (KPW), and the Lewisian Possible Worlds Account (LPW). This post will discuss what each view means, and where vagueness enters each account. I will also be identifying three crucial problems with (P) and showing how each of the other views answers these difficulties.
Here are brief definitions of each view, and how each one relates subjective degrees of rational confidence to probabilities (I will explain in more depth later).
Part of the reason for the previous confusion is that I was more or less assuming (P), and I think that Lauren had noticed some serious problems with it. First, a word on the reason for my assumption, and then I will try to state Lauren's objections.
(P) may be the dominant interpretation of Bayesianism. I don't really know. But there is good reason why someone reading the literature might think it's the dominant interpretation: it maps especially well to how Bayesian philosophers actually apply Bayesianism. Most philosophers who apply Bayesian reasoning (myself included) do it by simply making up numbers that are supposed to represent their degrees of confidence. Where do these numbers come from? We simply observe that we have varying degrees of confidence about different beliefs, and map these degrees of confidence to the real numbers between 0 and 1. Vagueness comes in from the fact that we don't have mathematically precise degrees of confidence, and our numbers are simply made up from our imprecise degrees of confidence, rather than computed somehow.
Now, as I have said, I believe that three important questions came out in our previous discussion which this theory leaves unanswered:
(P) does not answer these questions. This is, of course, to be expected from a theory called "primitivism," but I think the third question is particularly problematic. In the previous discussion, it was van Fraassen's assumption that we should do this very thing that brought up the issue. However, Bayesianism really needs these principles. Is it possible to provide an analysis of degrees of rational confidence that adequately answers these questions? (KPW) and (LPW) will attempt this very thing.
(KPW) is inspired by Kripke's treatment of possible worlds in terms of state spaces in the 1980 preface to Naming and Necessity, pp. 15-20. Kripke here argues that possible worlds are the same sorts of things as the "states" used in school probability theory, the difference being that possible worlds are maximally specific. Now, consider a view according to which the state space of Bayesian reasoning is the space of all epistemically possible worlds - that is, all the world-states (which are abstractions just like dice-states) which might for all we know be actual. Note that not all of these may be really possible. For instance, the Anselmian God either exists or does not exist, with logical necessity, but his existence and non-existence may both be epistemically possible for a particular person. So, when we say that we have subjective degree of confidence .5 for a given proposition, we are saying that that proposition holds in half of all epistemically possible worlds.
This view will be helped by Lewis's observation about the relationship between propositions and possible worlds: namely, that every proposition picks out a set of possible worlds, the worlds in which it obtains. (Lewis wants this to be a reductive analysis of propositions, but we need not do that.) So, consider any given proposition you believe. There is a set of possible worlds in which that proposition obtains. The set of epistemically possible worlds (for you) is the intersection of the sets for all the propositions you believe.
The (KPW) answer to question (1) has already been given - Bayesian degrees of confidence are probabilities. Let's proceed to give an interpretation of the math on (KPW).
Bayes' theorem is a relation between an initial probability - a probability over some state space S - and a conditional probability - a probability over some subset S' of S. Usually, we consider some proposition p and some evidence e. We already have assigned a particular degree of confidence to p and we want to adjust our confidence in light of learning the new evidence e. We use Bayes's theorem to calculate P(p|e). What has happened here? The new evidence e has eliminated certain formerly epistemically possible worlds - namely, all the worlds according to which ~e. In order to computer P(p|e) we have to know something about the relationship between p and e. In particular, we have to know P(e|p), P(p) and P(e) (all of them over the initial state space S). This involves knowing how many of our epistemically possible worlds certain conditions obtain in.
The (KPW) answer to question (3) should now be quite clear. Probabilities for events like dice rolls are, on this view, actually just special cases of degrees of subjective confidence. Why is there a 1/6 probability of a single die rolling 1? Because in 1/6 of all epistemically possible worlds it will land 1. (We should think of these world-states as covering the whole history of the world, so future events can be handled the same as past or present events.) In our school probability exercises, we simplify the case by supposing there are only 6 worlds. In fact, there are 6 sets of worlds. We know that the worlds will divide more or less evenly (we assume we know with certainty that the die will be rolled), because most of the propositions we are uncertain about vary independently of the result of the die roll. The ones that don't vary independently (e.g. propositions stating that the die is unfair in some particular way) are, for all we know, as likely to favor one side of the die as another.
Vagueness enters (KPW) by virtue of the fact that the worlds are created by us. They don't exist objectively. As such, there is vagueness as to how many worlds there are, and there is vagueness as to whether certain propositions are true in certain worlds. These things are simply not fully defined. We should nevertheless be able to fix upper and lower bounds by considering all of the possible resolutions of the vagueness (actually, we can probably do better by figuring out in advance which resolutions will lead to high values, and which to low values). In practice, however, we do something more like the die role case: we eliminate all the propositions that vary (more or less) independently (as far as we know) of the propositions under consideration, to divide the epistemically possible worlds into sets, and then consider each set as a single underspecified world.
(LPW) is very similar to (KPW) in its answers to the three questions. (LPW) holds that we are talking about real possible worlds, which are epistemically accessible - that is, which might, for all we know, be the world we're in. Vagueness on (LPW) is different. Because the worlds are fully defined and there is an objective truth about how many there are, there is only one source of vagueness properly so-called: vagueness about whether a given world is epistemically accessible. However, there is also second-order uncertainty - uncertainty about whether a certain world is genuinely possible, or whether a given proposition obtains in a certain world.
These two theories improve on (P) by providing explanations for why we use Bayesian reasoning the way we do, and why it works like probability theory at all. They also allow us to define our degrees of confidence much more clearly.
(Note: I tried to write this post last night and lost it when my powerbook overheated. Here goes the second time.)
David Lewis is best known for his modal realism, the view that all possible worlds exist in precisely the same sense that the actual world exists. He holds this view because he believes that it solves all sorts of philosophical problems related to modality, counterfactuals, properties, and so forth. However, there are a number of philosophers who think that the benefits of modal realism can be had without actually supposing that the possible world really exist. These philosophers Lewis calls ersatzers and in the section entitled "Paradise on the Cheap" in his book On the Plurality of Worlds Lewis attempts to reply to the ersatzers.
The first type of ersatzism to be dealt with is linguistic ersatzism. According to this view, possible worlds are linguistic constructs and therefore have only the ontological status of abstract objects like mathematical sets (whatever that might be) and not the status of concrete objects like the actual world. The linguistic ersatzer sets up a "world-making language" and asserts that possible worlds are maximal consistent sets of sentences in this language.
Lewis's first objection to this view (pp. 150-157) is that the ersatzer is required to assume certain facts about modality, which he is supposed to be explaining. In particular, Lewis wants to know whether it is possible for a particle to be both positively and negatively charged. Since these are (presumably) distinct predicates ("negatively charged" means more than just "not positively charged"), an axiom will be needed to enforce this rule (it's not a basic rule of the language). Lewis thinks that the ersatzer's axioms must come from some facts of modality distinct from his theory, so that his theory doesn't actually explain the facts of modality. In shot, Lewis claims that the ersatzer must be a primitivist about modality. To drive the point home, he wonders whether, according to the ersatzer, it is possible that there is a talking donkey. Again, Lewis says an axiom will be needed, and the axiom will depend on primitive facts of modality.
I suspect that Lewis is mistaken in his argument and that there is a reply open to the ersatzer based on distinguishing between different types of modality. By a "type of modality" I here mean a set of meanings that modal terms (e.g. "possible," "necessary," "impossible," etc.) can take in a certain context. Let's first distinguish between these types of modality and then consider the reply. The terms of modality can all be defined in terms of one another, so in my descriptions I will use whichever term is easiest to define.
In his discussion of possible worlds (and possible talking donkeys and positively and negatively charged particles) Lewis is talking about metaphysical or real modality. I've never heard a generic explanation of this type of modality that was more illuminating than its name, so suffice it to say that something is really possible just in case it really might have been actual, whatever really means. Lewis thinks this means that it really is actual for someone (the word "actual", according to Lewis, is an indexical like "here" or "now" - the actual world is just whatever world the speaker is in).
The next type of modality is narrowly logical or formal modality. A sentence is formally impossible relative to a language just in case the deductive calculus of that language can be used to derive an explicit contradiction (e.g. "A & ~A" in propositional logic) from it. Note that formal modal statements can be predicated of sentences, not of propositions and not of anything else. Also note that formal modal statements are always relative to a language.
The next concept is semantic or conceptual modality . A proposition is conceptually necessary just in case its truth is implicit in the definitions of the terms involved. For instance, it is conceptually necessary that a bachelor be an unmarried man (unless the context indicates that "bachelor" means "the recipient of a bachelor's degree" or something else, in which case it is not conceptually necessary).
Finally, there is broad logical modality. A proposition is broadly logically possible just in case all sentences that express it in some idealized language are formally possible relative to that language and it is conceptually possible. Now, of course, to give a full account of broad logical modality, you would have to give an account of the idealized language involved, but that's another story. Let's just suppose that there is some best formal logical language and we know what it is.
Now, Lewis seems to think, as I do, that real possibility and broad logical possibility are coextensive. Suppose the ersatzer also takes this view. Then what is the ersatzer doing? Well, he already has formal possibility by way of his language. The axioms Lewis wants him to add are the conceptually necessary truths. This isn't actually a problem for the ersatzer, because these are truths about language so they carry no additional commitments.
Considered from this direction, Lewis's objection seems a bit silly, especially when you consider that at the end he criticizes the ersatzer's introduction of axioms about talking-donkeyhood, saying "The job was to analyse modality ... It was not also part of the job to analyse 'talking donkey' (p. 156). The argument actually goes roughly like this (most of this is paraphrase of Lewis; the parts in italics I've added on behalf of the ersatzer):
LEWIS: Is it really possible that a single particle should be both positively and negatively charged?ERSATZER: I don't know. What do you mean by positive and negative charge?
LEWIS: It's your theory, so you tell me what positive and negative charge are.
ERSATZER: Well I don't know what positive and negative charge are, but if the definition of positive charge is such as to exclude its coexistence with negative charge in a single particle, then the answer to your question is yes. If the definition doesn't exclude this, then the answer is no.
LEWIS: Aren't you assuming facts about modality independent of your theory?
ERSATZER: No, I'm just assuming that terms like "positive charge" mean something, and that they mean the same thing when we're talking about modality as they do in the real world.
LEWIS: Well, then tell me this: is it possible that there should be a talking donkey?
ERSATZER: What do you mean by "talking donkey?"
LEWIS: You know, a donkey that talks!
ERSATZER: Well, it so happens that I know more about donkeys and talking than about positive and negative charge. A donkey is a certain arrangement of matter, and talking is a certain event having to do with vocal chords and sound waves, and these arrangements of matter are possible, so, yes, it is possible that there should be a talking donkey.
LEWIS: I asked you to analyze modality - why are you analyzing talking donkeys?
ERSATZER: How on earth am I supposed to tell you whether a talking donkey is possible without establishing what is meant by the words "talking donkey?"
Of course, this discussion is sympathetic to the ersatzer; from Lewis's perspective the objection is not so silly since I don't suppose he thinks these facts are just linguistic. Nevertheless, it seems to me that this reply on the part of the ersatzer is a simple and effective one.